If (x + 1/x)² = 3, then the value of (x^72 +x^66 + x^54+ x^24+ x^6 + 1) is
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\(x^{72} + x^{66} + x^{54} + x^{24} + x^{6} + 1 \in \left\{1 + \left(- \frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{66} + \left(- \frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{54} + \left(- \frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{24} + \left(- \frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{6} + \left(- \frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{72}, 1 + \left(- \frac{\sqrt{3}}{2} + \frac{i}{2}\right)^{72} + \left(- \frac{\sqrt{3}}{2} + \frac{i}{2}\right)^{6} + \left(- \frac{\sqrt{3}}{2} + \frac{i}{2}\right)^{54} + \left(- \frac{\sqrt{3}}{2} + \frac{i}{2}\right)^{24} + \left(- \frac{\sqrt{3}}{2} + \frac{i}{2}\right)^{66}, 1 + \left(\frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{66} + \left(\frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{6} + \left(\frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{54} + \left(\frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{24} + \left(\frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{72}, 1 + \left(\frac{\sqrt{3}}{2} + \frac{i}{2}\right)^{72} + \left(\frac{\sqrt{3}}{2} + \frac{i}{2}\right)^{54} + \left(\frac{\sqrt{3}}{2} + \frac{i}{2}\right)^{24} + \left(\frac{\sqrt{3}}{2} + \frac{i}{2}\right)^{6} + \left(\frac{\sqrt{3}}{2} + \frac{i}{2}\right)^{66}\right\}\)
\(i\) is the imaginary unit, defined as \(i^2 = -1\).
By substituting this value into the expression, we find that: $$\begin{aligned}x^{72} + x^{66} + x^{54} + x^{24} + x^{6} + 1 &= 1 + \left(- \frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{66} + \left(- \frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{54} + \left(- \frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{24} + \left(- \frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{6} + \left(- \frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{72}\\&= 1 + \left(- \frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{66} + \left(- \frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{54} + \left(- \frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{24} + \left(- \frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{6} + \left(- \frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{72}\end{aligned}$$